# inverse function theorem (topological spaces)

Let $X$ and $Y$ be topological spaces^{}, with $X$ compact^{} and $Y$ Hausdorff^{}. Suppose $f:X\to Y$ is a continuous^{} bijection. Then $f$ is a homeomorphism^{}, i.e. ${f}^{-1}$ is continuous.

Note if $Y$ is a metric space, then it is Hausdorff, and the theorem holds.

Title | inverse function theorem (topological spaces) |
---|---|

Canonical name | InverseFunctionTheoremtopologicalSpaces |

Date of creation | 2013-03-22 13:25:04 |

Last modified on | 2013-03-22 13:25:04 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 54C05 |

Related topic | Compact |